Question

What is the standard deviation of the random variable x "the number that comes up on the die"?

Answer 1

Answer:2.0

Explanation:

Just did it on a p e x

Answer 2

Assuming a fair 6-sided die, the random variable
`X` giving the number that comes up follows a uniform distribution with PMF

`P(X=x)=\begin{cases}\frac16&\text{for }x\in\{1,2,3,4,5,6\}\\\\0&\text{otherwise}\end{cases}`

The standard deviation of
`X` is the square root of the variance of
`X`,
`V[X]`. We have a formula for the variance in terms of the expected value,
`E[X]`:

`V[X]=E[X^2]-E[X]^2`

where

`E[X]=\displaystyle\sum_xx\,P(X=x)=\sum_(x=1)^6\frac x6=\frac72`

`E[X^2]=\displaystyle\sum_xx^2\,P(X=x)=\sum_(x=1)^6\frac{x^2}6=\frac{91}6`

Then the variance is

`V[X]=\frac{91}6-\left(\frac72\right)^2=(35)/(12)`

so the standard deviation is

`√(V[X])=\sqrt{(35)/(12)}\approx1.71`